Average Error: 22.8 → 0.1
Time: 6.8s
Precision: binary64
\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -119727822.23598845 \lor \neg \left(y \leq 195088.09069458197\right):\\ \;\;\;\;\left(x + \left(\frac{1}{y} + \frac{x}{y \cdot y}\right)\right) - \left(\frac{x}{y} + \frac{1}{y \cdot y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{1 + {y}^{3}} \cdot \left(y \cdot y + \left(1 - y\right)\right)\\ \end{array}\]
1 - \frac{\left(1 - x\right) \cdot y}{y + 1}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -119727822.23598845 \lor \neg \left(y \leq 195088.09069458197\right):\\
\;\;\;\;\left(x + \left(\frac{1}{y} + \frac{x}{y \cdot y}\right)\right) - \left(\frac{x}{y} + \frac{1}{y \cdot y}\right)\\

\mathbf{else}:\\
\;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{1 + {y}^{3}} \cdot \left(y \cdot y + \left(1 - y\right)\right)\\

\end{array}
(FPCore (x y) :precision binary64 (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))
↓
(FPCore (x y)
 :precision binary64
 (if (or (<= y -119727822.23598845) (not (<= y 195088.09069458197)))
   (- (+ x (+ (/ 1.0 y) (/ x (* y y)))) (+ (/ x y) (/ 1.0 (* y y))))
   (- 1.0 (* (/ (* y (- 1.0 x)) (+ 1.0 (pow y 3.0))) (+ (* y y) (- 1.0 y))))))
double code(double x, double y) {
	return 1.0 - (((1.0 - x) * y) / (y + 1.0));
}
↓
double code(double x, double y) {
	double tmp;
	if ((y <= -119727822.23598845) || !(y <= 195088.09069458197)) {
		tmp = (x + ((1.0 / y) + (x / (y * y)))) - ((x / y) + (1.0 / (y * y)));
	} else {
		tmp = 1.0 - (((y * (1.0 - x)) / (1.0 + pow(y, 3.0))) * ((y * y) + (1.0 - y)));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original22.8
Target0.3
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y < -3693.8482788297247:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y < 6799310503.41891:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -119727822.235988453 or 195088.09069458197 < y

    1. Initial program 45.6

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(x + \left(\frac{x}{{y}^{2}} + \frac{1}{y}\right)\right) - \left(\frac{x}{y} + \frac{1}{{y}^{2}}\right)}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\left(x + \left(\frac{1}{y} + \frac{x}{y \cdot y}\right)\right) - \left(\frac{x}{y} + \frac{1}{y \cdot y}\right)}\]

    if -119727822.235988453 < y < 195088.09069458197

    1. Initial program 0.1

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
    2. Using strategy rm

    3. Applied flip3-+_binary64_167900.1

      \[\leadsto 1 - \frac{\left(1 - x\right) \cdot y}{\color{blue}{\frac{{y}^{3} + {1}^{3}}{y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)}}}\]

    4. Applied associate-/r/_binary64_167330.1

      \[\leadsto 1 - \color{blue}{\frac{\left(1 - x\right) \cdot y}{{y}^{3} + {1}^{3}} \cdot \left(y \cdot y + \left(1 \cdot 1 - y \cdot 1\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -119727822.23598845 \lor \neg \left(y \leq 195088.09069458197\right):\\ \;\;\;\;\left(x + \left(\frac{1}{y} + \frac{x}{y \cdot y}\right)\right) - \left(\frac{x}{y} + \frac{1}{y \cdot y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{1 + {y}^{3}} \cdot \left(y \cdot y + \left(1 - y\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021043 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1.0 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))) (- (/ 1.0 y) (- (/ x y) x))))

  (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))